# Estimation of market price of risk of short interest rate under the Hull-White model

I think I am a bit confused. I intend to estimate the market price of risk the short interest rate, say, under the Hull-White model. I have the following two questions.

1. Is it correct to state state that the (zero coupon) bond price and the (zero coupon) bond European option price are both valued in the risk-neutral measure, because they are both derivatives of the short rate? If so, the parameters calibrated from these prices would not provide the market price of risk since the latter could only be estimated in the real world measure. Is this correct?

2. If the answer to the above question is affirmative, how does one estimate the market price of risk (risk premium) of the short interest rate?

• +1. Thank you. Regarding your answer #2, suppose the short rate follows the Hull-White model, what determines the realized bond excess return and how is it related to the short rate risk premium? The Hull-White model implies the bond price $P(t,T)$ follows $\frac{dP(t,T)}{P(t,T)}=r(t)\,dt−\sigma_0 B(t,T)\,dW_t$. So the return is just the short interest rate $r$. Is there an extra term to $r$ to result in the excess return? – Hans Aug 16 '18 at 17:51
• Here is the correction to the confusing remark in my last comment. $dP=rPdt+\sigma\frac{\partial P}{\partial r}dW_Q=\Big(rP+\lambda\sigma\frac{\partial P}{\partial r}\Big)dt+\sigma\frac{\partial P}{\partial r}dW$, where $W$ and $W_Q$ stand for the Brownian motions in the physical measure and the risk neutral measure $Q$, respectively, and $\sigma$ for the instantaneous volatility of the short rate $r$, $\lambda$ the interest rate risk premium. So $\lambda\sigma\frac1P\frac{\partial P}{\partial r}$ is the excess return in the physical measure. – Hans Aug 26 '18 at 3:35